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Using the computer in space geometryL.M. Doorman, H v.d. Kooij

A previous version of this article appeared in Zentralblatt für Didaktik der Mathemtik, Jahrgang 24, Heft 5, 1992.

Abstract

First some changes in the space geometry curriculum in The Netherlandssketched. A part of the changes in the lower grades of secondary education amost the complete change of the space geometry in the higher grades. Durinsketch the need for the use of a computer will raise. In the second part of this pit will be shown how computertools assist this new space geometry curriculum

1 Space geometry

1.1 The national program

In 1990 space geometry regained its place as a main subject in the curriculusecondary education in The Netherlands. Until that year students in higher gradpre-vocational education had to solve problems on three dimensional geometryalgebraic tools. Most of the problems posed in final examinations couldn’t evesolved in a geometrical way, because they were not designed for such treatmeIn the new curriculum vectoralgebra has disappeared. It is no longer possible tvectoralgebra for solving an exercise, without having any idea of the spatial situaConsequently the students will be confronted with problems in which “they havuse their spatial imaginative power to find a solution” (quoted from the new NatioProgram). Much attention must be paid to the development of students’ spatialitions not only as a preparation for space geometry, but mainly as a source for getry in general.The nationwide introduction of the new program was preceded by the Hawex-pr(1987 - 1991) for pre-vocational education (age 15 - 17). Through field tests thecurriculum was developed. Some of the schools also used computertools to suspace geometry.Space-geometry for the lower grades (age 12 - 16) is changed through the pW12-16. This new curriculum has been tested in a number of schools. The nawide introduction of this part of the curriculum took place in august 1993.

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1.2 Some space-geometry in the lower grades

Among others there are two important goals for space-geometry in the lower gr(see [E. de Moor]):• Students learn to "read" pictures of spatial situations, like photo’s, maps, etc.

must be able to make a spatial image of the picture and make drawings of thage on paper.

• Students must be able to see intersecting planes in spatial objects and try to por to draw the true (two-dimensional) shapes of these planes.

In the first two years of secondary education students learn how a spatial effect intures of for example a cube is created (see fig. 1 below). Students work with csections of spatial objects. This prepares for situations in every day life where mproperties of spatial objects are explained by their cross-sections (e.g. in architebiology, medicine).In the higher grades students have to work with pictures of spatial situations anjects. A problem for these students is that in the past very little attention has beento space geometry in the lower grades. Especially to the ability: "How to read pictof three-dimensional situations?" Therefore in the higher grades students oftenot able to make the correct spatial images of these pictures. For example:

Figure 1: Do these two lines intersect?

The answer of a student (age 16) in the question above: "Intersecting, because thelay in one plane (RPSQ)."How to start this part of space-geometry? According to didactical principles ofreal-istic mathematics education[1] the space around us, is very often used as a sourceproblem posing. So you start with real (or mathematical) objects and pictures thawell known to the students. What spatial objects do you intersect in every day

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Food! An example is the following exercise about a large sausage:

Figure 2: How to cut a sausage?

> Are the cross-sections possible? If yes, explain how to cut the sausage in ormake the cross-section visible.

With this kind of questioning we try to reach the above mentioned goals. By confring the students alternately with three-dimensional objects, with pictures of thremensional objects and cross-sections in these objects we practice the ability ofing spatial views.From food the step is made to more mathematical figures, like the cube and theamid. When you make this step you don’t want to bother students with difficult cstruction methods. These methods can be avoided by using a computer. The com(with the appropriate software) can create a fluent layer between concrete matethat have to be used in space-geometry, and pictures or drawings. Pictures ocomputerscreen are not just two-dimensional pictures. When you are able to mulate an object, to rotate or enlarge for example, you get a better idea of the waobject is situated in space. So an important feature of the software should be:

To create an environment in which students can investigate cross-sections in podra, cut these polyhedra into parts, and can simply correct wrong cuttings.

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1.3 Space-geometry in the higher grades

1.3.1 What do you see?

The following example is taken from one of the Hawex bookletsTekenen wat je weet(Draw what you know) [2].

Figure 3: Cube and shadow.

This photograph of a cube was taken on a tropical island, so the sunbeams areed almost perpendicular to the ground.> In the picture only part of the shadow is visible. What will the shadow look li

“behind” the cube?By changing the position of the cube or the sun, you can find different shapes of sow. Assume the edge of the cube has length 1.> At which positions of sun and cube the shadow will be a square of 1 by 1?> Describe a situation in which the shadow is a square with edges of length

These are difficult problems. Without the help of concrete materials most stud(and teachers?) cannot solve it. Having real cubes by hand students work (ingroups) on this problem for a long time, very actively examining the cubes frmany different viewpoints. At the end of the course students must be able to imamentallywhat certain objects look like from any given viewpoint. For many studethis is very difficult to learn. You need some courage to do so. Very often you hthe lamentation in the classroom “I don’t see it!” Of course concrete (models of)ids are very important for the students to get familiar to the idea of how objectslike from any viewpoint.We felt that a certain kind of software should also be of great help as an intermephase between the use of concrete materials and the mental activities we wanteto perform. A first important feature of this software is:

To offer the possibility to look at objects from different viewpoints.

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Such software is also helpful for problems like:

Figure 4: Six times two lines

In this picture you see two linesl andm in a top-view projection (l → l’ , m → m’).

> Which positions of the two lines (intersecting, skew, parallel) are possible in eof the six pictures?

In general one picture is not enough to decide about the positions of the two lTwo pictures from different viewpoints may give enough information about thesitions.

1.3.2 Dynamical definitions of spatial objects

Look at the following definition of a prism, taken from a textbook on geometry:“A prism is a solid composed of a number of faces, the upper and the lower faceparallel and the lateral faces are parallelograms.”It is difficult to imagine some real solid by such a dull, static and meaninglessscription. With their hands moving through the air, but mentally too, studentsbuild solids very easily with the help of so calleddynamical definitions (or recipes).The recipe for a prism: “A prism arises by moving a polygon parallel to the startsition along a given line.”Once you have chosen a polygon and a direction for the translation, you cangrow. For defining a pyramid in this way, you only have to reduce linearly the pereter of the polygon.The choice for these dynamic definitions is a didactical one. It offers the studentpossibility to “grasp the space”. Immaterial objects like the sunbeams and the wing space of robotarms become almost material too when described dynamica

Figure 5: Working space of robotarms.

> What shape has the workspace of each of the robots?

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The enclosed space of an electronic wave guide can be created by translatingthe same time rotating a rectangle along a given line.

Figure 6: Electronic wave guide.

> Will this inner space fit exactly in a cylinder?> What is the shape of the inner opening?

Again a software tool can be very helpful when you are working on problemsthese. A second important feature of the software should be:

To offer the possibility to translate, rotate and enlarge/reduce two- and three-dimsional objects.

Figure 7: Dynamic representations of a cone, a prism and a pyramid

> Describe the arising of the cone out of a circle.> The same question when a triangle is used as a starting point.

1.3.3 Two dimensional representations of spatial objects

Two quotations from the National Program:“Concrete examples (like sculptures and buildings) are very important as illustions for students to get insight into the subjects of space geometry. Photographdrawings can be used presenting such objects.”“The student must know that every method of drawing is a convention and tharelation to the three-dimensional reality depends on the method of drawing thused.”

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For these reasons attention is paid to different methods of projection. In the followfigure you see two ways of drawing a cube: an example of an oblique projectionone of an orthogonal projection. Both are parallel projections.

Figure 8: Two projections of a cube

The left one is commonly used in textbooks on geometry. Many students and teadon’t recognize a cube in the right picture, because they are so used to the lefDo they recognize a cube, or do they recognize the picture? The same projemethods used for a cone will show that the oblique projection doesn’t give a naperspective.

Figure 9: Two projections of a cone

Using photographs and drawings as a source for problems means that studentto learn about differences between methods of projection. And not only about theallel projections, but also about the central projection.

oblique orthogonal

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Architects for example, mostly use an orthogonal parallel projection when tpresent their creations:

Figure 10: Drawing from an architect.

But often they use an oblique parallel projection, for example to make a planroom look more spatial:

Figure 11: View of a room.

When students use photographs they must know the difference between parallcentral projection. Using the computer the difference can be examined by puttineye near to the object (central) or far away from the object (parallel and orthogo

Figure 12: A cube

> What is the position of the eye (related to the cube) when you see the cubeway?

When the above picture is a front-view of the cube, a side-view is very helpfusolve the problem:

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Figure 13: A side-view of the projection of a cube

Very specific ways of orthogonal parallel projection are the XY-projection (toview), the XZ-projection (side-view) and the YZ-projection (front-view). These thways of drawing are often very useful in exploring problems. Front-view and tview are also easy tools to investigate the next problem:

Figure 14: An exercise.

The faces DCF and ABE are parallel. Lines AB and CD are skew. In the pictureseems to be the shortest horizontal connection between AB and CD.

> Find the horizontal connection between AB and CD that really is the shortest

The last important feature for the software that we defined is:

To offer the possibility to use different methods of projection and to compare th

1.4 Summary

For the features mentioned above software was needed in both the project Wand the Hawex-project. At that moment nothing was available that could meet oumands. So the msdos-computerprograms Doorzien and Ruimfig [3] were deveduring the projects. During this development parts of the software were adaptthe curriculum and parts of the curriculum were adapted to the software. Sschools involved in the projects worked with Doorzien or Ruimfig. Doorzien is atranslated into German with all kinds of studentmaterials and didactical hints [4].two programs are in the mean time redesigned into one program Doorzien fordows.

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2 The computerWe take the computertool Doorzien as an example of using the computer in spacometry. Problems are posed “outside” the programs and can be explored usinsoftware. In the first paragraph 2.1 an idea of the use of Doorzien in the lower gris given. The next three paragraphs will show how the program is used in thespace geometry curriculum for the higher grades. Each paragraph contains onefeatures the software was developed for.

2.1 Intersecting a cube

After some introductory exercises two girls in the second grade (age±14) arrive atthe cube:

Figure 15: Draw a cross-section in the cube and draw the shape of the cross-section next t

They draw the plane through two parallel face-diagonals in the cube. This crosstion is familiar for them. A rectangle is drawn next to it. It seems as if there is no prlem. But with the next cross-section, they drew a parallellogram:

Figure 16: What is the shape of this cross-section?

Strange. Self made cross-sections were no problem for the girls, but the shapecross-section is wrong. In the next exercise they discover a feature of the compprogram. This feature takes the cross-section out of the cube and rotates it untsee the real shape (i.e. looking at it from a perpendicular direction).

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After giving the proper command they see on the screen:

Figure 17: The shape of the cross-section

"Oh...of course...", one of the girls mumbles, "a cube has only right angles"!? Shmoves the first solution of the exercise and draws the rectangle next to the cubAnother exercise is about a cube with a triangular cross-section. With the compprogram it is possible to move this cross-section in the cube. The movement ispendicular to the cross-section, the next cross-section is parallel to the previousWhat different shapes have the parallel cross-sections?The girls move the triangle carefully backwards, until they see the largest possibangle:

Figure 18: Largest triangle in the cube

They were asked what would happen when the cross-section moves a littlebackwards. "Well, you pass this point." She points to a vertex of the cube. Sheplains with help from the picture on the screen that the cross-section will get oneside passing that point. She concludes: "There are three such points, so the crotion will have six sides when you move it backwards." They move the cross-secbackwards and see their hypothesis come true.

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Doorzien offers more possibilities for investigating a cross-section. A polyhedcan be cut in two parts. Each of the two parts can be selected for further investigaAt any moment it is possible to let the computer draw a fold-out of the polyhed

Figure 19: A fold-out (Schermafdruk?)

In this way students can create their own diamond-like polyhedra, print the fold-omake glue strips and put them together. Unfortunately, no printer was available iclassroom at that moment.While the two girls worked with Doorzien, they had to make spatial images of ptures on the screen. This training of spatial ability is an important part of space geetry. At the same time the exercises and the way students work with Doorzien megoals that were formulated in the first part of this paper.

2.2 Projecting the cube

After exploring spatial objects and their cross-section spatial coordinates and dent projections are introduced. The standard projection of a cube in Doorzien iengineers-projection. When also the axes are draw you get the following pictur

Figure 20: A cube in Doorzien

The position of the cube in space is not clear from this picture. It is possible to chother projections. The students (16 years old) are asked to look at the three pa

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projections of the cube on the XY-plane, the YZ-plane and the XZ-plane. Unitsvisible on the axes.

Figure 21: The projection on the XZ-plane

> What are the coordinates of the corner points of the cube?> Move the centre of the cube to the origin (using the submenu “Transform”).> Check your transformation with the projections on the different planes.

When the central projection is used, you can change the point from which it prothe object on a plane through the origin, perpendicular to the line from this pointhe origin. The last question involves this method of projection.

> From what position the central projection gives the same picture as the projeon the YZ-plane?

With these exercises students get familiar with Doorzien and with some of the adtages and disadvantages of the program. For example it is hard to see whethera line is in front of another line. In general, one projection of an object doesn’t genough information about the position of the object in space. While answering thercises the students get a notion of this principle in an “active way”. They learndifferent projections give different information about the object. You need the tand the side-views for the determination of the coordinates and for measurinsides of the cube, but other projections, like the central projection from a proper vpoint, give a better spatial view of the object.

2.3 The electronic wave guide

The exercise about the electronic wave guide is from one of the booklets oHawex-project and is mentioned before. The electronic wave guide can be bufrom a rectangle with the command “Translate and Rotate”. When the object isished it is easy to project it in different ways. Doing this, some questions abouwave guide can be answered easily.

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First we need a rectangle. Objects can be built and after successive combinedformations Translate and Rotate you see the following:

Figure 22: An electronic waveguide in Doorzien

During the transformation the waveguide arises on the screen as a sequence ofand translated rectangles. The arising gives a spatial effect to the picture . Thinamical effect can not be illustrated on paper. Now you can visualize more diffiquestions about this wave guide.

> What function describes the route of a vertex in top- and side-views?

The parallel projection in the direction of the translation reveals the next picturanswers the previous question in the booklet about the shape of the openingwave guide.

Figure 23: Front-view of the electronic waveguide

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2.4 Art in Eindhoven

This problem deals with the facility of the program to look at an object from differviewpoints using the central projection. The exercise starts with three photograpa sculpture in Eindhoven.

Figure 24: Three photographs of the object.

First the students have to make this object in Doorzien.

Figure 25: The sculpture (not finished yet) in a cube in Doorzien.

Second, they are asked questions about this object.

> From what position the object looks like:

Figure 26: Different views of the sculpture

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When students solve this problem they carefully move their head around an inviobject in their hands. Trying to look at the “real” object from the right position.Such exercises are perfect as an intermediate stage between working with comaterials and working with pictures or photographs in space geometry. Especwhen there are still some concrete materials near the computers that can be uthe students.

3 Final remarksIt is difficult to give an accurate description of the contribution of the computerspace geometry. To what extend does it help in developing some spatial imaginpower?With the examples in this paper we showed that using software gives at least(didactical) possibilities in space geometry. With a last example we will showalso students learn something from the software.In the worksheets that support Doorzien students had to answer questions aboutviews of a cube. The front-view (a square) was on the screen and they were asfind rotations to get given other front-views of the cube, for example a hexagonOne of the teachers asked the students a few lessons after working with DoorzA front-view of a cube is a square. The cube can be rotated around a vertical axeyou see the following picture:

Figure 27: A picture of a cube

> Around how many degrees the cube is rotated?

Most students came up with the following drawing:

Figure 28: Drawing a top-view

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When this step is made by the students, we think that working with softwareDoorzien really helped them in space geometry. The main reason for using theputer is not in the first place the pictures that can be made easily. The main purpare the features mentioned in chapter two:1. To create an environment in which students can investigate cross-sections i

yhedra, cut these polyhedra into parts, and easily can correct wrong cutting2. To offer the possibility for looking at three dimensional situations from differe

viewpoints in order to collect more information about the situation.3. To offer the possibility to translate, rotate and enlarge/reduce two- and thre

mensional objects. In such a way that three-dimensional objects can be builtnamically” from two-dimensional objects.

4. To offer the possibility to use different methods of projection and to compthem.

notes

1 For more information on realistic mathematics education we refer to:- A. Treffers, F. Goffree:Rational Analysis of Realistic Mathematics Education - TheWiskobas Program, Proceedings of the Ninth International Conference for the Psychologyof Mathematics Education Vol II, Freudenthal Institute, The Netherlands, 1985.- J. de Lange:Mathematics, Insight and Meaning, Freudenthal Institute, The Netherlands,1987.- K. Gravemeier e.a.:Context Free Productions Tests and Geometry in Realistic Mathemat-ics Education, Freudenthal Institute, The Netherlands, 1990.

More about space geometry in The Netherlands can be found in:- A. Goddijn, M. Kindt:Space Geometry Doesn’t Fit in the Book, Proceedings of the NinthInternational Conference for the Psychology of Mathematics Education Vol I, FreudenthalInstitute, The Netherlands, 1985.- E. de Moor:Geometry-instruction (age 4-14) in The Netherlands -the realistic approach-, Realistic Mathematics Education in Primary School (ed. L. Streefland), Freudenthal Insti-tute, The Netherlands, 1991.

2 Materials on space geometry developed at the Freudenthal institute are:- A. Goddijn:Shadow and depth, 1980.- M. Kindt, J. de Lange:Lessons in space geometry, 1982.- A. Roodhardt:Verkenning in de ruimte, 1989.- A. Roodhardt:Tekenen wat je weet, 1990.- A. Roodhardt:Op maat gesneden, 1990.

3 For information about Ruimfig or Doorzien please contact: L.M. Doorman(M.Doorman@fi.uu.nl). A demo-version of a new program for Windows in which Ruimfigand Doorzien are integrated is available on this cdrom (see the TWIN-project for a link tothe application).

4 Heinz Schumann:Korperschnitte, Raumgeometrie interaktiv mit dem computer. Dümmler,1995.

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